Method and apparatus for controlling a non-linear mill

ABSTRACT

A method and apparatus for controlling a non-linear mill. A linear controller is provided having a linear gain k that is operable to receive inputs representing measured variables of the plant and predict on an output of the linear controller predicted control values for manipulatible variables that control the plant. A non-linear model of the plant is provided for storing a representation of the plant over a trained region of the operating input space and having a steady-state gain K associated therewith. The gain k of the linear model is adjusted with the gain K of the non-linear model in accordance with a predetermined relationship as the measured variables change the operating region of the input space at which the linear controller is predicting the values for the manipulatible variables. The predicted manipulatible variables are then output after the step of adjusting the gain k

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is a Continuation of U.S. patent applicationSer. No. 09/514,733 (Atty Dkt. No. PAVI-24,956) entitled METHOD ANDAPPARATUS FOR CONTROLLING A NON-LINEAR MILL,” which is aContinuation-in-Part of issued U.S. Pat. No. 6,487,459, issued Nov. 26,2002 entitled “METHOD AND APPARATUS FOR MODELING DYNAMIC AND STEADYSTATE PROCESSES FOR PREDICTION, CONTROL AND OPTIMIZATION,” which is acontinuation of issued U.S. Pat. No. 5,933,345, issued Aug. 3, 1999,entitled “METHOD AND APPARATUS FOR DYNAMIC AND STEADY STATE MODELINGOVER A DESIRED PATH BETWEEN TWO END POINTS.”

TECHNICAL FIELD OF THE INVENTION

[0002] This invention pertains in general to modeling techniques and,more particularly, to combining steady-state and dynamic models for thepurpose of prediction, control and optimization for non-linear millcontrol.

BACKGROUND OF THE INVENTION

[0003] Process models that are utilized for prediction, control andoptimization can be divided into two general categories, steady-statemodels and dynamic models. In each case the model is a mathematicalconstruct that characterizes the process, and process measurements areutilized to parameterize or fit the model so that it replicates thebehavior of the process. The mathematical model can then be implementedin a simulator for prediction or inverted by an optimization algorithmfor control or optimization.

[0004] Steady-state or static models are utilized in modem processcontrol systems that usually store a great deal of data, this datatypically containing steady-state information at many differentoperating conditions. The steady-state information is utilized to traina non-linear model wherein the process input variables are representedby the vector U that is processed through the model to output thedependent variable Y. The non-linear model is a steady-statephenomenological or empirical model developed utilizing several orderedpairs (U_(i), Y_(i)) of data from different measured steady states. If amodel is represented as:

Y=P(U, Y)  (001)

[0005] where P is some parameterization, then the steady-state modelingprocedure can be presented as:

({right arrow over (U)},{right arrow over (Y)})→P  (002)

[0006] where U and Y are vectors containing the U_(i), Y_(i) orderedpair elements. Given the model P, then the steady-state process gain canbe calculated as: $\begin{matrix}{K = \frac{\Delta \quad {P\left( {U,Y} \right)}}{\Delta \quad U}} & (003)\end{matrix}$

[0007] The steady-state model therefore represents the processmeasurements that are taken when the system is in a “static” mode. Thesemeasurements do not account for the perturbations that exist whenchanging from one steady-state condition to another steady-statecondition. This is referred to as the dynamic part of a model.

[0008] A dynamic model is typically a linear model and is obtained fromprocess measurements which are not steady-state measurements; rather,these are the data obtained when the process is moved from onesteady-state condition to another steady-state condition. This procedureis where a process input or manipulated variable u(t) is input to aprocess with a process output or controlled variable y(t) being outputand measured. Again, ordered pairs of measured data (u(I), y(I)) can beutilized to parameterize a phenomenological or empirical model, thistime the data coming from non-steady-state operation. The dynamic modelis represented as:

y(t)=p(u(t),y(t))  (004)

[0009] where p is some parameterization. Then the dynamic modelingprocedure can be represented as:

({right arrow over (u)}, {right arrow over (y)})→p  (005)

[0010] Where u and y are vectors containing the (u(I),y(I)) ordered pairelements. Given the model p, then the steady-state gain of a dynamicmodel can be calculated as: $\begin{matrix}{k = \frac{\Delta \quad {p\left( {u,y} \right)}}{\Delta \quad u}} & (006)\end{matrix}$

[0011] Unfortunately, almost always the dynamic gain k does not equalthe steady-state gain K, since the steady-state gain is modeled on amuch larger set of data, whereas the dynamic gain is defined around aset of operating conditions wherein an existing set of operatingconditions are mildly perturbed. This results in a shortage ofsufficient non-linear information in the dynamic data set in whichnon-linear information is contained within the static model. Therefore,the gain of the system may not be adequately modeled for an existing setof steady-state operating conditions. Thus, when considering twoindependent models, one for the steady-state model and one for thedynamic model, there is a mis-match between the gains of the two modelswhen used for prediction, control and optimization. The reason for thismis-match are that the steady-state model is non-linear and the dynamicmodel is linear, such that the gain of the steady-state model changesdepending on the process operating point, with the gain of the linearmodel being fixed. Also, the data utilized to parameterize the dynamicmodel do not represent the complete operating range of the process,i.e., the dynamic data is only valid in a narrow region. Further, thedynamic model represents the acceleration properties of the process(like inertia) whereas the steady-state model represents the tradeoffsthat determine the process final resting value (similar to the tradeoffbetween gravity and drag that determines terminal velocity in freefall).

[0012] One technique for combining non-linear static models and lineardynamic models is referred to as the Hammerstein model. The Hammersteinmodel is basically an input-output representation that is decomposedinto two coupled parts. This utilizes a set of intermediate variablesthat are determined by the static models which are then utilized toconstruct the dynamic model. These two models are not independent andare relatively complex to create.

SUMMARY OF THE INVENTION

[0013] The present invention disclosed and claimed herein comprises amethod for controlling a non-linear plant. A linear controller isprovided having a linear gain k that is operable to receive inputsrepresenting measured variables of the plant and predict on an output ofthe linear controller predicted control values for manipulatiblevariables that control the plant. A non-linear model of the plant isprovided for storing a representation of the plant over a trained regionof the operating input space and having a steady-state gain K associatedtherewith. The gain k of the linear model is adjusted with the gain K ofthe non-linear model in accordance with a predetermined relationship asthe measured variables change the operating region of the input space atwhich the linear controller is predicting the values for themanipulatible variables. The predicted manipulatible variables are thenoutput after the step of adjusting the gain k.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] For a more complete understanding of the present invention andthe advantages thereof, reference is now made to the followingdescription taken in conjunction with the accompanying Drawings inwhich:

[0015]FIG. 1 illustrates a prior art Hammerstein model;

[0016]FIG. 2 illustrates a block diagram of the modeling technique ofthe present invention;

[0017]FIGS. 3a-3 d illustrate timing diagrams for the various outputs ofthe system of FIG. 2;

[0018]FIG. 4 illustrates a detailed block diagram of the dynamic modelutilizing the identification method;

[0019]FIG. 5 illustrates a block diagram of the operation of the modelof FIG. 4;

[0020]FIG. 6 illustrates an example of the modeling technique of thepresent invention utilized in a control environment;

[0021]FIG. 7 illustrates a diagrammatic view of a change between twosteady-state values;

[0022]FIG. 8 illustrates a diagrammatic view of the approximationalgorithm for changes in the steady-state value;

[0023]FIG. 9 illustrates a block diagram of the dynamic model;

[0024]FIG. 10 illustrates a detail of the control network utilizing theerror constraining algorithm of the present invention;

[0025]FIGS. 11a and 11 b illustrate plots of the input and output duringoptimization;

[0026]FIG. 12 illustrates a plot depicting desired and predictedbehavior;

[0027]FIG. 13 illustrates various plots for controlling a system toforce the predicted behavior to the desired behavior;

[0028]FIG. 14 illustrates a plot of the trajectory weighting algorithmof the present invention;

[0029]FIG. 15 illustrates a plot for the constraining algorithm;

[0030]FIG. 16 illustrates a plot of the error algorithm as a function oftime;

[0031]FIG. 17 illustrates a flowchart depicting the statistical methodfor generating the filter and defining the end point for theconstraining algorithm of FIG. 15;

[0032]FIG. 18 illustrates a diagrammatic view of the optimizationprocess;

[0033]FIG. 18a illustrates a diagrammatic representation of the mannerin which the path between steady-state values is mapped through theinput and output space;

[0034]FIG. 19 illustrates a flowchart for the optimization procedure;

[0035]FIG. 20 illustrates a diagrammatic view of the input space and theerror associated therewith;

[0036]FIG. 21 illustrates a diagrammatic view of the confidence factorin the input space;

[0037]FIG. 22 illustrates a block diagram of the method for utilizing acombination of a non-linear system and a first principal system; and

[0038]FIG. 23 illustrates an alternate embodiment of the embodiment ofFIG. 22.

[0039]FIG. 24 illustrates a block diagram of a kiln, cooler andpreheater;

[0040]FIG. 25 illustrates a block diagram of a non-linear controlledmill;

[0041]FIG. 26 illustrates a table for various aspects of the millassociated with the fresh feed and the separator speed;

[0042]FIG. 27 illustrates historical modeling data for the mill;

[0043]FIGS. 28a and 28 b illustrate diagrams of sensitivity versus thepercent of the Blaine and the percent of the Return, respectively;

[0044]FIG. 29 illustrates non-linear gains for fresh feed responses;

[0045]FIG. 30 illustrates plots of non-linear gains for separator speedresponses; and

[0046]FIG. 31 illustrates a closed loop non-linear model predictivecontrol, target change for Blaine.

DETAILED DESCRIPTION OF THE INVENTION

[0047] Referring now to FIG. 1, there is illustrated a diagrammatic viewof a Hammerstein model of the prior art. This is comprised of anon-linear static operator model 10 and a linear dynamic model 12, bothdisposed in a series configuration. The operation of this model isdescribed in H. T. Su, and T. J. McAvoy, “Integration of MultilayerPerceptron Networks and Linear Dynamic Models: A Hammerstein ModelingApproach” to appear in I & EC Fundamentals, paper dated Jul. 7, 1992,which reference is incorporated herein by reference. Hammerstein modelsin general have been utilized in modeling non-linear systems for sometime. The structure of the Hammerstein model illustrated in FIG. 1utilizes the non-linear static operator model 10 to transform the inputU into intermediate variables H. The non-linear operator is usuallyrepresented by a finite polynomial expansion. However, this couldutilize a neural network or any type of compatible modeling system. Thelinear dynamic operator model 12 could utilize a discreet dynamictransfer function representing the dynamic relationship between theintermediate variable H and the output Y. For multiple input systems,the non-linear operator could utilize a multilayer neural network,whereas the linear operator could utilize a two layer neural network. Aneural network for the static operator is generally well known anddescribed in U.S. Pat. No. 5,353,207, issued Oct. 4, 1994, and assignedto the present assignee, which is incorporated herein by reference.These type of networks are typically referred to as a multilayerfeed-forward network which utilizes training in the form ofback-propagation. This is typically performed on a large set of trainingdata. Once trained, the network has weights associated therewith, whichare stored in a separate database.

[0048] Once the steady-state model is obtained, one can then choose theoutput vector from the hidden layer in the neural network as theintermediate variable for the Hammerstein model. In order to determinethe input for the linear dynamic operator, u(t), it is necessary toscale the output vector h(d) from the non-linear static operator model10 for the mapping of the intermediate variable h(t) to the outputvariable of the dynamic model y(t), which is determined by the lineardynamic model.

[0049] During the development of a linear dynamic model to represent thelinear dynamic operator, in the Hammerstein model, it is important thatthe steady-state non-linearity remain the same. To achieve this goal,one must train the dynamic model subject to a constraint so that thenon-linearity learned by the steady-state model remains unchanged afterthe training. This results in a dependency of the two models on eachother.

[0050] Referring now to FIG. 2, there is illustrated a block diagram ofthe modeling method of the present invention, which is referred to as asystematic modeling technique. The general concept of the systematicmodeling technique in the present invention results from the observationthat, while process gains (steady-state behavior) vary with U's andY's,(i.e., the gains are non-linear), the process dynamics seeminglyvary with time only, (i.e., they can be modeled as locally linear, buttime-varied). By utilizing non-linear models for the steady-statebehavior and linear models for the dynamic behavior, several practicaladvantages result. They are as follows:

[0051] 1. Completely rigorous models can be utilized for thesteady-state part. This provides a credible basis for economicoptimization.

[0052] 2. The linear models for the dynamic part can be updated on-line,i.e., the dynamic parameters that are known to be time-varying can beadapted slowly.

[0053] 3. The gains of the dynamic models and the gains of thesteady-state models can be forced to be consistent (k=K).

[0054] With further reference to FIG. 2, there are provided a static orsteady-state model 20 and a dynamic model 22. The static model 20, asdescribed above, is a rigorous model that is trained on a large set ofsteady-state data. The static model 20 will receive a process input Uand provide a predicted output Y. These are essentially steady-statevalues. The steady-state values at a given time are latched in variouslatches, an input latch 24 and an output latch 26. The latch 24 containsthe steady-state value of the input U_(ss), and the latch 26 containsthe steady-state output value Y_(ss). The dynamic model 22 is utilizedto predict the behavior of the plant when a change is made from asteady-state value of Y_(ss) to a new value Y. The dynamic model 22receives on the input the dynamic input value u and outputs a predicteddynamic value y. The value u is comprised of the difference between thenew value U and the steady-state value in the latch 24, U_(ss). This isderived from a subtraction circuit 30 which receives on the positiveinput thereof the output of the latch 24 and on the negative inputthereof the new value of U. This therefore represents the delta changefrom the steady-state. Similarly, on the output the predicted overalldynamic value will be the sum of the output value of the dynamic model,y, and the steady-state output value stored in the latch 26, Y_(ss).These two values are summed with a summing block 34 to provide apredicted output Y. The difference between the value output by thesumming junction 34 and the predicted value output by the static model20 is that the predicted value output by the summing junction 20accounts for the dynamic operation of the system during a change. Forexample, to process the input values that are in the input vector U bythe static model 20, the rigorous model, can take significantly moretime than running a relatively simple dynamic model. The method utilizedin the present invention is to force the gain of the dynamic model 22k_(d) to equal the gain K_(ss) of the static model 20.

[0055] In the static model 20, there is provided a storage block 36which contains the static coefficients associated with the static model20 and also the associated gain value K_(ss). Similarly, the dynamicmodel 22 has a storage area 38 that is operable to contain the dynamiccoefficients and the gain value k_(d). One of the important aspects ofthe present invention is a link block 40 that is operable to modify thecoefficients in the storage area 38 to force the value of k_(d) to beequal to the value of K_(ss). Additionally, there is an approximationblock 41 that allows approximation of the dynamic gain k_(d) between themodification updates.

[0056] Systematic Model

[0057] The linear dynamic model 22 can generally be represented by thefollowing equations: $\begin{matrix}{{\delta \quad {y(t)}} = {{\sum\limits_{i = 1}^{n}\quad {b_{i}\delta \quad {u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}\quad {a_{i}\delta \quad {y\left( {t - i} \right)}}}}} & (007)\end{matrix}$

[0058] where:

δy(t)=y(t)−Y _(ss)  (008)

δu(t)=u(t)−u _(ss)  (009)

[0059] and t is time, a_(i) and b_(i) are real numbers, d is a timedelay, u(t) is an input and y(t) an output. The gain is represented by:$\begin{matrix}{\frac{y(B)}{u(B)} = {k = \frac{\left( {\sum\limits_{i = 1}^{n}\quad {b_{i}B^{i - 1}}} \right)B^{d}}{1 + {\sum\limits_{i = 1}^{n}\quad {a_{i}B^{i - 1}}}}}} & (10)\end{matrix}$

[0060] where B is the backward shift operator B(x(t))=x(t−1), t=time,the a_(i) and b_(i) are real numbers, I is the number of discreet timeintervals in the dead-time of the process, and n is the order of themodel. This is a general representation of a linear dynamic model, ascontained in George E. P. Box and G. M Jenkins, “TIME SERIES ANALYSISforecasting and control”, Holden-Day, San Francisco, 1976, Section 10.2,Page 345. This reference is incorporated herein by reference.

[0061] The gain of this model can be calculated by setting the value ofB equal to a value of “1”. The gain will then be defined by thefollowing equation: $\begin{matrix}{\left\lbrack \frac{y(B)}{u(B)} \right\rbrack_{B = 1} = {k_{d} = \frac{\sum\limits_{i = 1}^{n}\quad b_{i}}{1 + {\sum\limits_{i = 1}^{n}\quad a_{i}}}}} & (11)\end{matrix}$

[0062] The a_(i) contain the dynamic signature of the process, itsunforced, natural response characteristic. They are independent of theprocess gain. The b_(i) contain part of the dynamic signature of theprocess; however, they alone contain the result of the forced response.The b_(i) determine the gain k of the dynamic model. See: J. L. Shearer,A. T. Murphy, and H. H. Richardson, “Introduction to System Dynamics”,Addison-Wesley, Reading, Massachusetts, 1967, Chapter 12. This referenceis incorporated herein by reference.

[0063] Since the gain K_(ss) of the steady-state model is known, thegain k_(d) of the dynamic model can be forced to match the gain of thesteady-state model by scaling the b_(i) parameters. The values of thestatic and dynamic gains are set equal with the value of b_(i) scaled bythe ratio of the two gains: $\begin{matrix}{\left( b_{i} \right)_{scaled} = {\left( b_{i} \right)_{old}\left( \frac{K_{ss}}{k_{d}} \right)}} & (12) \\{\left( b_{i} \right)_{scaled} = \frac{\left( b_{i} \right)_{old}{K_{ss}\left( {1 + {\sum\limits_{i = 1}^{n}\quad a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}\quad b_{i}}} & (13)\end{matrix}$

[0064] This makes the dynamic model consistent with its steady-statecounterpart. Therefore, each time the steady-state value changes, thiscorresponds to a gain K_(ss) of the steady-state model. This value canthen be utilized to update the gain k_(d) of the dynamic model and,therefore, compensate for the errors associated with the dynamic modelwherein the value of k_(d) is determined based on perturbations in theplant on a given set of operating conditions. Since all operatingconditions are not modeled, the step of varying the gain will accountfor changes in the steady-state starting points.

[0065] Referring now to FIGS. 3a-3 d, there are illustrated plots of thesystem operating in response to a step function wherein the input valueU changes from a value of 100 to a value of 110. In FIG. 3a, the valueof 100 is referred to as the previous steady-state value U_(ss). In FIG.3b, the value of u varies from a value of 0 to a value of 10, thisrepresenting the delta between the steady-state value of U_(ss) to thelevel of 110, represented by reference numeral 42 in FIG. 3a. Therefore,in FIG. 3b the value of u will go from 0 at a level 44, to a value of 10at a level 46. In FIG. 3c, the output Y is represented as having asteady-state value Y_(ss) of 4 at a level 48. When the input value Urises to the level 42 with a value of 110, the output value will rise.This is a predicted value. The predicted value which is the properoutput value is represented by a level 50, which level 50 is at a valueof 5. Since the steady-state value is at a value of 4, this means thatthe dynamic system must predict a difference of a value of 1. This isrepresented by FIG. 3d wherein the dynamic output value y varies from alevel 54 having a value of 0 to a level 56 having a value of 1.0.However, without the gain scaling, the dynamic model could, by way ofexample, predict a value for y of 1.5, represented by dashed level 58,if the steady-state values were outside of the range in which thedynamic model was trained. This would correspond to a value of 5.5 at alevel 60 in the plot of FIG. 3c. It can be seen that the dynamic modelmerely predicts the behavior of the plant from a starting point to astopping point, not taking into consideration the steady-state values.It assumes that the steady-state values are those that it was trainedupon. If the gain k_(d) were not scaled, then the dynamic model wouldassume that the steady-state values at the starting point were the samethat it was trained upon. However, the gain scaling link between thesteady-state model and the dynamic model allow the gain to be scaled andthe parameter b_(i) to be scaled such that the dynamic operation isscaled and a more accurate prediction is made which accounts for thedynamic properties of the system.

[0066] Referring now to FIG. 4, there is illustrated a block diagram ofa method for determining the parameters a_(i), b_(i). This is usuallyachieved through the use of an identification algorithm, which isconventional. This utilizes the (u(t),y(t)) pairs to obtain the a_(i)and b_(i) parameters. In the preferred embodiment, a recursiveidentification method is utilized where the a_(i) and b_(i) parametersare updated with each new (u_(i)(t),y_(i)(t)) pair. See: T. Eykhoff,“System Identification”, John Wiley & Sons, New York, 1974, Pages 38 and39, et. seq., and H. Kurz and W. Godecke, “Digital Parameter-AdaptiveControl Processes with Unknown Dead Time”, Automatica, Vol. 17, No. 1,1981, pp. 245-252, which references are incorporated herein byreference.

[0067] In the technique of FIG. 4, the dynamic model 22 has the outputthereof input to a parameter-adaptive control algorithm block 60 whichadjusts the parameters in the coefficient storage block 38, which alsoreceives the scaled values of k, b_(i). This is a system that is updatedon a periodic basis, as defined by timing block 62. The controlalgorithm 60 utilizes both the input u and the output y for the purposeof determining and updating the parameters in the storage area 38.

[0068] Referring now to FIG. 5, there is illustrated a block diagram ofthe preferred method. The program is initiated in a block 68 and thenproceeds to a function block 70 to update the parameters a_(i), b_(i)utilizing the (u(I),y(I)) pairs. Once these are updated, the programflows to a function block 72 wherein the steady-state gain factor K isreceived, and then to a function block 74 to set the dynamic gain to thesteady state gain, i.e., provide the scaling function describedhereinabove. This is performed after the update. This procedure can beused for on-line identification, non-linear dynamic model prediction andadaptive control.

[0069] Referring now to FIG. 6, there is illustrated a block diagram ofone application of the present invention utilizing a controlenvironment. A plant 78 is provided which receives input values u(t) andoutputs an output vector y(t). The plant 78 also has measurable statevariables s(t). A predictive model 80 is provided which receives theinput values u(t) and the state variables s(t) in addition to the outputvalue y(t). The steady-state model 80 is operable to output a predictedvalue of both y(t) and also of a future input value u(t+1). Thisconstitutes a steady-state portion of the system. The predictedsteady-state input value is U_(ss) with the predicted steady-stateoutput value being Y_(ss). In a conventional control scenario, thesteady-state model 80 would receive as an external input a desired valueof the output y^(d)(t) which is the desired value that the overallcontrol system seeks to achieve. This is achieved by controlling adistributed control system (DCS) 86 to produce a desired input to theplant. This is referred to as u(t+1), a future value. Withoutconsidering the dynamic response, the predictive model 80, asteady-state model, will provide the steady-state values. However, whena change is desired, this change will effectively be viewed as a “stepresponse”.

[0070] To facilitate the dynamic control aspect, a dynamic controller 82is provided which is operable to receive the input u(t), the outputvalue y(t) and also the steady-state values U_(ss) and Y_(ss) andgenerate the output u(t+1). The dynamic controller effectively generatesthe dynamic response between the changes, i.e., when the steady-statevalue changes from an initial steady-state value U_(ss) ^(i), Y_(ss)^(i) to a final steady-state value U^(f) _(ss), Y^(f) _(ss).

[0071] During the operation of the system, the dynamic controller 82 isoperable in accordance with the embodiment of FIG. 2 to update thedynamic parameters of the dynamic controller 82 in a block 88 with again link block 90, which utilizes the value K_(ss) from a steady-stateparameter block in order to scale the parameters utilized by the dynamiccontroller 82, again in accordance with the above described method. Inthis manner, the control function can be realized. In addition, thedynamic controller 82 has the operation thereof optimized such that thepath traveled between the initial and final steady-state values isachieved with the use of the optimizer 83 in view of optimizerconstraints in a block 85. In general, the predicted model (steady-statemodel) 80 provides a control network function that is operable topredict the future input values. Without the dynamic controller 82, thisis a conventional control network which is generally described in U.S.Pat. No. 5,353,207, issued Oct. 4, 1994, to the present assignee, whichpatent is incorporated herein by reference.

[0072] Approximate Systematic Modeling

[0073] For the modeling techniques described thus far, consistencybetween the steady-state and dynamic models is maintained by rescalingthe b_(i) parameters at each time step utilizing equation 13. If thesystematic model is to be utilized in a Model Predictive Control (MPC)algorithm, maintaining consistency may be computationally expensive.These types of algorithms are described in CE. Garcia, D. M. Prett andM. Morari. Model predictive control: theory and practice—a survey,Automatica, 25:335-348, 1989; D. E. Seborg, T. F. Edgar, and D. A.Mellichamp. Process Dynamics and Control. John Wiley and Sons, New York,N.Y., 1989. These references are incorporated herein by reference. Forexample, if the dynamic gain k_(d) is computed from a neural networksteady-state model, it would be necessary to execute the neural networkmodule each time the model was iterated in the MPC algorithm. Due to thepotentially large number of model iterations for certain MPC problems,it could be computationally expensive to maintain a consistent model. Inthis case, it would be better to use an approximate model which does notrely on enforcing consistencies at each iteration of the model.

[0074] Referring now to FIG. 7, there is illustrated a diagram for achange between steady state values. As illustrated, the steady-statemodel will make a change from a steady-state value at a line 100 to asteady-state value at a line 102. A transition between the twosteady-state values can result in unknown settings. The only way toinsure that the settings for the dynamic model between the twosteady-state values, an initial steady-state value K_(ss) ^(i) and afinal steady-state gain K_(ss) ^(f), would be to utilize a stepoperation, wherein the dynamic gain k_(d) was adjusted at multiplepositions during the change. However, this may be computationallyexpensive. As will be described hereinbelow, an approximation algorithmis utilized for approximating the dynamic behavior between the twosteady-state values utilizing a quadratic relationship. This is definedas a behavior line 104, which is disposed between an envelope 106, whichbehavior line 104 will be described hereinbelow.

[0075] Referring now to FIG. 8, there is illustrated a diagrammatic viewof the system undergoing numerous changes in steady-state value asrepresented by a stepped line 108. The stepped line 108 is seen to varyfrom a first steady-state value at a level 110 to a value at a level 112and then down to a value at a level 114, up to a value at a level 116and then down to a final value at a level 118. Each of these transitionscan result in unknown states. With the approximation algorithm that willbe described hereinbelow, it can be seen that, when a transition is madefrom level 110 to level 112, an approximation curve for the dynamicbehavior 120 is provided. When making a transition from level 114 tolevel 116, an approximation gain curve 124 is provided to approximatethe steady state gains between the two levels 114 and 116. For makingthe transition from level 116 to level 118, an approximation gain curve126 for the steady-state gain is provided. It can therefore be seen thatthe approximation curves 120-126 account for transitions betweensteady-state values that are determined by the network, it being notedthat these are approximations which primarily maintain the steady-stategain within some type of error envelope, the envelope 106 in FIG. 7.

[0076] The approximation is provided by the block 41 noted in FIG. 2 andcan be designed upon a number of criteria, depending upon the problemthat it will be utilized to solve. The system in the preferredembodiment, which is only one example, is designed to satisfy thefollowing criteria:

[0077] 1. Computational Complexity: The approximate systematic modelwill be used in a Model Predictive Control algorithm, therefore, it isrequired to have low computational complexity.

[0078] 2. Localized Accuracy: The steady-state model is accurate inlocalized regions. These regions represent the steady-state operatingregimes of the process. The steady-state model is significantly lessaccurate outside these localized regions.

[0079] 3. Final Steady-State: Given a steady-state set point change, anoptimization algorithm which uses the steady-state model will be used tocompute the steady-state inputs required to achieve the set point.Because of item 2, it is assumed that the initial and finalsteady-states associated with a set-point change are located in regionsaccurately modeled by the steady-state model.

[0080] Given the noted criteria, an approximate systematic model can beconstructed by enforcing consistency of the steady-state and dynamicmodel at the initial and final steady-state associated with a set pointchange and utilizing a linear approximation at points in between the twosteady-states. This approximation guarantees that the model is accuratein regions where the steady-state model is well known and utilizes alinear approximation in regions where the steady-state model is known tobe less accurate. In addition, the resulting model has low computationalcomplexity. For purposes of this proof, Equation 13 is modified asfollows: $\begin{matrix}{b_{i,{scaled}} = \frac{b_{i}{K_{ss}\left( {u\left( {t - d - 1} \right)} \right)}\left( {1 + {\sum\limits_{i = 1}^{n}\quad a_{i}}} \right)}{\sum\limits_{i = 1}^{n}\quad b_{i}}} & (14)\end{matrix}$

[0081] This new equation 14 utilizes K_(ss)(u(t−d−1)) instead ofK_(ss)(u(t)) as the consistent gain, resulting in a systematic modelwhich is delay invariant.

[0082] The approximate systematic model is based upon utilizing thegains associated with the initial and final steady-state values of aset-point change. The initial steady-state gain is denoted K^(i) _(ss)while the initial steady-state input is given by U^(i) _(ss). The finalsteady-state gain is K^(f) _(ss) and the final input is U^(f) _(ss).Given these values, a linear approximation to the gain is given by:$\begin{matrix}{{K_{ss}\left( {u(t)} \right)} = {K_{ss}^{i} + {\frac{K_{ss}^{f} - K_{ss}^{i}}{U_{ss}^{f} - U_{ss}^{i}}{\left( {{u(t)} - U_{ss}^{i}} \right).}}}} & (15)\end{matrix}$

[0083] Substituting this approximation into Equation 13 and replacingu(t−d−1)−u^(i) by δu(t−d−1) yields: $\begin{matrix}{{\overset{\sim}{b}}_{j,{scaled}} = {\frac{b_{j}{K_{ss}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}\quad a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}\quad b_{i}} + {\frac{1}{2}\frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}\quad a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}\quad b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}\delta \quad {{u\left( {t - d - i} \right)}.}}}} & (16)\end{matrix}$

[0084] To simplify the expression, define the variable b_(j)-Bar as:$\begin{matrix}{{\overset{\_}{b}}_{j} = \frac{b_{j}{K_{ss}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}\quad a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}\quad b_{i}}} & (17)\end{matrix}$

[0085] and g_(j) as: $\begin{matrix}{g_{j} = \frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}\quad a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}\quad b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}} & (18)\end{matrix}$

[0086] Equation 16 may be written as:

{tilde over (b)} _(j,scaled) ={overscore (b)} _(j) +g _(j)δu(t−d−i)  (19)

[0087] Finally, substituting the scaled b's back into the originaldifference Equation 7, the following expression for the approximatesystematic model is obtained: $\begin{matrix}{{\delta \quad {y(t)}} = {{\sum\limits_{i = 1}^{n}\quad {{\overset{\_}{b}}_{i}\delta \quad {u\left( {t - d - i} \right)}}} + {\sum\limits_{i = 1}^{n}\quad {g_{i}\delta \quad {u\left( {t - d - i^{2}} \right)}\delta \quad {u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}\quad {a_{i}\delta \quad {y\left( {t - i} \right)}}}}} & (20)\end{matrix}$

[0088] The linear approximation for gain results in a quadraticdifference equation for the output. Given Equation 20, the approximatesystematic model is shown to be of low computational complexity. It maybe used in a MPC algorithm to efficiently compute the required controlmoves for a transition from one steady-state to another after aset-point change. Note that this applies to the dynamic gain variationsbetween steady-state transitions and not to the actual path values.

[0089] Control System Error Constraints

[0090] Referring now to FIG. 9, there is illustrated a block diagram ofthe predictin engine for the dynamic controller 82 of FIG. 6. Theprediction engine is operable to essentially predict a value of y(t) asthe predicted future value y(t+1). Since the prediction engine mustdetermine what the value of the output y(t) is at each future valuebetween two steady-state values, it is necessary to perform these in a“step” manner. Therefore, there will be k steps from a value of zero toa value of N, which value at k=N is the value at the “horizon”, thedesired value. This, as will be described hereinbelow, is an iterativeprocess, it being noted that the terminology for “(t+1)” refers to anincremental step, with an incremental step for the dynamic controllerbeing smaller than an incremented step for the steady-state model. Forthe steady-state model, “y(t+N)” for the dynamic model will be, “y(t+1)”for the steady state The value y(t+1) is defined as follows:

y(t+1)=a ₁ y(t)+a ₂ y(t−1)+b ₁ u(t−d−1)+b ₂ u(t−d−2)  (021)

[0091] With further reference to FIG. 9, the input values u(t) for each(u,y) pair are input to a delay line 140. The output of the delay lineprovides the input value u(t) delayed by a delay value “d”. There areprovided only two operations for multiplication with the coefficients b₁and b₂, such that only two values u(t) and u(t−1) are required. Theseare both delayed and then multiplied by the coefficients b₁ and b₂ andthen input to a summing block 141. Similarly, the output value y^(p)(t)is input to a delay line 142, there being two values required formultiplication with the coefficients a₁ and a₂. The output of thismultiplication is then input to the summing block 141. The input to thedelay line 142 is either the actual input value y^(a)(t) or the iteratedoutput value of the summation block 141, which is the previous valuecomputed by the dynamic controller 82. Therefore, the summing block 141will output the predicted value y(t+1) which will then be input to amultiplexor 144. The multiplexor 144 is operable to select the actualoutput y^(a)(t) on the first operation and, thereafter, select theoutput of the summing block 141. Therefore, for a step value of k=0 thevalue y^(a)(t) will be selected by the multiplexor 144 and will belatched in a latch 145. The latch 145 will provide the predicted valuey^(p)(t+k) on an output 146. This is the predicted value of y(t) for agiven k that is input back to the input of delay line 142 formultiplication with the coefficients a₁ and a₂. This is iterated foreach value of k from k=0 to k=N.

[0092] The a₁ and a₂ values are fixed, as described above, with the b₁and b₂ values scaled. This scaling operation is performed by thecoefficient modification block 38. However, this only defines thebeginning steady-state value and the final steady-state value, with thedynamic controller and the optimization routines described in thepresent application defining how the dynamic controller operates betweenthe steady-state values and also what the gain of the dynamic controlleris. The gain specifically is what determines the modification operationperformed by the coefficient modification block 38.

[0093] In FIG. 9, the coefficients in the coefficient modification block38 are modified as described hereinabove with the information that isderived from the steady-state model. The steady-state model is operatedin a control application, and is comprised in part of a forwardsteady-state model 141 which is operable to receive the steady-stateinput value U_(ss)(t) and predict the steady-state output valueY_(ss)(t). This predicted value is utilized in an inverse steady-statemodel 143 to receive the desired value y^(d)(t) and the predicted outputof the steady-state model 141 and predict a future steady-state inputvalue or manipulated value U_(ss)(t+N) and also a future steady-stateinput value Y_(ss)(t+N) in addition to providing the steady-state gainK_(ss). As described hereinabove, these are utilized to generate scaledb-values. These b-values are utilized to define the gain k_(d) of thedynamic model. In can therefore be seen that this essentially takes alinear dynamic model with a fixed gain and allows it to have a gainthereof modified by a non-linear model as the operating point is movedthrough the output space.

[0094] Referring now to FIG. 10, there is illustrated a block diagram ofthe dynamic controller and optimizer. The dynamic controller includes adynamic model 149 which basically defines the predicted value y^(p)(k)as a function of the inputs y(t), s(t) and u(t). This was essentiallythe same model that was described hereinabove with reference to FIG. 9.The model 149 predicts the output values y^(p)(k) between the twosteady-state values, as will be described hereinbelow. The model 149 ispredefined and utilizes an identification algorithm to identify the a₁,a₂, b₁ and b₂ coefficients during training. Once these are identified ina training and identification procedure, these are “fixed”. However, asdescribed hereinabove, the gain of the dynamic model is modified byscaling the coefficients b₁ and b₂. This gain scaling is not describedwith respect to the optimization operation of FIG. 10, although it canbe incorporated in the optimization operation.

[0095] The output of model 149 is input to the negative input of asumming block 150. Summing block 150 sums the predicted output y^(p)(k)with the desired output y^(d)(t). In effect, the desired value ofy^(d)(t) is effectively the desired steady-state value Y^(f) _(ss),although it can be any desired value. The output of the summing block150 comprises an error value which is essentially the difference betweenthe desired value y^(d)(t) and the predicted value y^(p)(k). The errorvalue is modified by an error modification block 151, as will bedescribed hereinbelow, in accordance with error modification parametersin a block 152. The modified error value is then input to an inversemodel 153, which basically performs an optimization routine to predict achange in the input value u(t). In effect, the optimizer 153 is utilizedin conjunction with the model 149 to minimize the error output bysumming block 150. Any optimization function can be utilized, such as aMonte Carlo procedure. However, in the present invention, a gradientcalculation is utilized. In the gradient method, the gradient ∂(y)/∂(u)is calculated and then a gradient solution performed as follows:$\begin{matrix}{{\Delta \quad u_{new}} = {{\Delta \quad u_{old}} + {\left( \frac{\partial(y)}{\partial(u)} \right) \times E}}} & (022)\end{matrix}$

[0096] The optimization function is performed by the inverse model 153in accordance with optimization constraints in a block 154. An iterationprocedure is performed with an iterate block 155 which is operable toperform an iteration with the combination of the inverse model 153 andthe predictive model 149 and output on an output line 156 the futurevalue u(t+k+1). For k=0, this will be the initial steady-state value andfor k=N, this will be the value at the horizon, or at the nextsteady-state value. During the iteration procedure, the previous valueof u(t+k) has the change value Au added thereto. This value is utilizedfor that value of k until the error is within the appropriate levels.Once it is at the appropriate level, the next u(t+k) is input to themodel 149 and the value thereof optimized with the iterate block 155.Once the iteration procedure is done, it is latched. As will bedescribed hereinbelow, this is a combination of modifying the error suchthat the actual error output by the block 150 is not utilized by theoptimizer 153 but, rather, a modified error is utilized. Alternatively,different optimization constraints can be utilized, which are generatedby the block 154, these being described hereinbelow.

[0097] Referring now to FIGS. 11a and 11 b, there are illustrated plotsof the output y(t+k) and the input u_(k)(t+k+1), for each k from theinitial steady-state value to the horizon steady-state value at k=N.With specific reference to FIG. 11a, it can be seen that theoptimization procedure is performed utilizing multiple passes. In thefirst pass, the actual value u^(a)(t+k) for each k is utilized todetermine the values of y(t+k) for each u,y pair. This is thenaccumulated and the values processed through the inverse model 153 andthe iterate block 155 to minimize the error. This generates a new set ofinputs u_(k)(t+k+1) illustrated in FIG. 11b. Therefore, the optimizationafter pass 1 generates the values of u(t+k+1) for the second pass. Inthe second pass, the values are again optimized in accordance with thevarious constraints to again generate another set of values foru(t+k+1). This continues until the overall objective function isreached. This objective function is a combination of the operations as afunction of the error and the operations as a function of theconstraints, wherein the optimization constraints may control theoverall operation of the inverse model 153 or the error modificationparameters in block 152 may control the overall operation. Each of theoptimization constraints will be described in more detail hereinbelow.

[0098] Referring now to FIG. 12, there is illustrated a plot of y^(d)(t)and y^(p)(t). The predicted value is represented by a waveform 170 andthe desired output is represented by a waveform 172, both plotted overthe horizon between an initial steady-state value Y^(i) _(ss) and afinal steady-state value Y^(f) _(ss). It can be seen that the desiredwaveform prior to k=0 is substantially equal to the predicted output. Atk=0, the desired output waveform 172 raises its level, thus creating anerror. It can be seen that at k=0, the error is large and the systemthen must adjust the manipulated variables to minimize the error andforce the predicted value to the desired value. The objective functionfor the calculation of error is of the form: $\begin{matrix}{\min\limits_{\Delta \quad u_{il}}{\sum\limits_{j}{\sum\limits_{k}\quad \left( {A_{j}*\left( {{{\overset{\rightarrow}{y}}^{p}(t)} - {{\overset{\rightarrow}{y}}^{d}(t)}} \right)^{2}} \right.}}} & (23)\end{matrix}$

[0099] where:

[0100] Du_(il) is the change in input variable (IV) I at time interval 1

[0101] A_(j) is the weight factor for control variable (CV) j

[0102] y^(p)(t) is the predicted value of CV j at time interval k

[0103] y^(d)(t) is the desired value of CV j.

[0104] Trajectory Weighting

[0105] The present system utilizes what is referred to as “trajectoryweighting” which encompasses the concept that one does not put aconstant degree of importance on the future predicted process behaviormatching the desired behavior at every future time set, i.e., at lowk-values. One approach could be that one is more tolerant of error inthe near term (low k-values) than farther into the future (highk-values). The basis for this logic is that the final desired behavioris more important than the path taken to arrive at the desired behavior,otherwise the path traversed would be a step function. This isillustrated in FIG. 13 wherein three possible predicted behaviors areillustrated, one represented by a curve 174 which is acceptable, one isrepresented by a different curve 176, which is also acceptable and onerepresented by a curve 178, which is unacceptable since it goes abovethe desired level on curve 172. Curves 174-178 define the desiredbehavior over the horizon for k=1 to N.

[0106] In Equation 23, the predicted curves 174-178 would be achieved byforcing the weighting factors A_(j) to be time varying. This isillustrated in FIG. 14. In FIG. 14, the weighting factor A as a functionof time is shown to have an increasing value as time and the value of kincreases. This results in the errors at the beginning of the horizon(low k-values) being weighted much less than the errors at the end ofthe horizon (high k-values). The result is more significant than merelyredistributing the weights out to the end of the control horizon at k=N.This method also adds robustness, or the ability to handle a mismatchbetween the process and the prediction model. Since the largest error isusually experienced at the beginning of the horizon, the largest changesin the independent variables will also occur at this point. If there isa mismatch between the process and the prediction (model error), theseinitial moves will be large and somewhat incorrect, which can cause poorperformance and eventually instability. By utilizing the trajectoryweighting method, the errors at the beginning of the horizon areweighted less, resulting in smaller changes in the independent variablesand, thus, more robustness.

[0107] Error Constraints

[0108] Referring now to FIG. 15, there are illustrated constraints thatcan be placed upon the error. There is illustrated a predicted curve 180and a desired curve 182, desired curve 182 essentially being a flatline. It is desirable for the error between curve 180 and 182 to beminimized. Whenever a transient occurs at t=0, changes of some sort willbe required. It can be seen that prior to t=0, curve 182 and 180 aresubstantially the same, there being very little error between the two.However, after some type of transition, the error will increase. If arigid solution were utilized, the system would immediately respond tothis large error and attempt to reduce it in as short a time aspossible. However, a constraint frustum boundary 184 is provided whichallows the error to be large at t=0 and reduces it to a minimum level ata point 186. At point 186, this is the minimum error, which can be setto zero or to a non-zero value, corresponding to the noise level of theoutput variable to be controlled. This therefore encompasses the sameconcepts as the trajectory weighting method in that final futurebehavior is considered more important that near term behavior. The evershrinking minimum and/or maximum bounds converge from a slack positionat t=0 to the actual final desired behavior at a point 186 in theconstraint frustum method.

[0109] The difference between constraint frustums and trajectoryweighting is that constraint frustums are an absolute limit (hardconstraint) where any behavior satisfying the limit is just asacceptable as any other behavior that also satisfies the limit.Trajectory weighting is a method where differing behaviors havegraduated importance in time. It can be seen that the constraintsprovided by the technique of FIG. 15 requires that the value y^(p)(t) isprevented from exceeding the constraint value. Therefore, if thedifference between y^(d)(t) and y^(p)(t) is greater than that defined bythe constraint boundary, then the optimization routine will force theinput values to a value that will result in the error being less thanthe constraint value. In effect, this is a “clamp” on the differencebetween y^(p)(t) and y^(d)(t). In the trajectory weighting method, thereis no “clamp” on the difference therebetween; rather, there is merely anattenuation factor placed on the error before input to the optimizationnetwork.

[0110] Trajectory weighting can be compared with other methods, therebeing two methods that will be described herein, the dynamic matrixcontrol (DMC) algorithm and the identification and command (IdCom)algorithm. The DMC algorithm utilizes an optimization to solve thecontrol problem by minimizing the objective function: $\begin{matrix}{\min\limits_{\Delta \quad U_{il}}{\underset{j\quad}{\sum\quad}\underset{k\quad}{\sum\quad}\left( {{A_{j}*\left( {{{\overset{\rightarrow}{y}}^{P}(t)} - {{\overset{\rightarrow}{y}}^{D}(t)}} \right)} + {\underset{i\quad}{\sum\quad}B_{i}*{\sum\limits_{1}\quad \left( {\Delta \quad U_{il}} \right)^{2}}}} \right.}} & (24)\end{matrix}$

[0111] where B_(i) is the move suppression factor for input variable I.This is described in Cutler, C. R. and B. L. Ramaker, Dynamic MatrixControl—A Computer Control Algorithm, AIChE National Meeting, Houston,Tex. (April, 1979), which is incorporated herein by reference.

[0112] It is noted that the weights A_(j) and desired values y^(d)(t)are constant for each of the control variables. As can be seen fromEquation 24, the optimization is a trade off between minimizing errorsbetween the control variables and their desired values and minimizingthe changes in the independent variables. Without the move suppressionterm, the independent variable changes resulting from the set pointchanges would be quite large due to the sudden and immediate errorbetween the predicted and desired values. Move suppression limits theindependent variable changes, but for all circumstances, not just theinitial errors.

[0113] The IdCom algorithm utilizes a different approach. Instead of aconstant desired value, a path is defined for the control variables totake from the current value to the desired value. This is illustrated inFIG. 16. This path is a more gradual transition from one operation pointto the next. Nevertheless, it is still a rigidly defined path that mustbe met. The objective function for this algorithm takes the form:$\begin{matrix}{\min\limits_{\Delta \quad U_{il}}{\sum\limits_{j}{\sum\limits_{k}\quad \left( {A_{j}*\left( {Y^{P_{jk}} - y_{refjk}} \right)} \right)^{2}}}} & (25)\end{matrix}$

[0114] This technique is described in Richalet, J., A. Rault, J. L.Testud, and J. Papon, Model Predictive Heuristic Control: Applicationsto Industrial Processes, Automatica, 14, 413-428 (1978), which isincorporated herein by reference. It should be noted that therequirement of Equation 25 at each time interval is sometimes difficult.In fact, for control variables that behave similarly, this can result inquite erratic independent variable changes due to the control algorithmattempting to endlessly meet the desired path exactly.

[0115] Control algorithms such as the DMC algorithm that utilize a formof matrix inversion in the control calculation, cannot handle controlvariable hard constraints directly. They must treat them separately,usually in the form of a steady-state linear program. Because this isdone as a steady-state problem, the constraints are time invariant bydefinition. Moreover, since the constraints are not part of a controlcalculation, there is no protection against the controller violating thehard constraints in the transient while satisfying them at steady-state.

[0116] With further reference to FIG. 15, the boundaries at the end ofthe envelope can be defined as described hereinbelow. One techniquedescribed in the prior art, W. Edwards Deming, “Out of the Crisis,”Massachusetts Institute of Technology, Center for Advanced EngineeringStudy, Cambridge Mass., Fifth Printing, September 1988, pages 327-329,describes various Monte Carlo experiments that set forth the premisethat any control actions taken to correct for common process variationactually may have a negative impact, which action may work to increasevariability rather than the desired effect of reducing variation of thecontrolled processes. Given that any process has an inherent accuracy,there should be no basis to make a change based on a difference thatlies within the accuracy limits of the system utilized to control it. Atpresent, commercial controllers fail to recognize the fact that changesare undesirable, and continually adjust the process, treating alldeviation from target, no matter how small, as a special cause deservingof control actions, i.e., they respond to even minimal changes. Overadjustment of the manipulated variables therefore will result, andincrease undesirable process variation. By placing limits on the errorwith the present filtering algorithms described herein, only controlleractions that are proven to be necessary are allowed, and thus, theprocess can settle into a reduced variation free from unmeritedcontroller disturbances. The following discussion will deal with onetechnique for doing this, this being based on statistical parameters.

[0117] Filters can be created that prevent model-based controllers fromtaking any action in the case where the difference between thecontrolled variable measurement and the desired target value are notsignificant. The significance level is defined by the accuracy of themodel upon which the controller is statistically based. This accuracy isdetermined as a function of the standard deviation of the error and apredetermined confidence level. The confidence level is based upon theaccuracy of the training. Since most training sets for a neuralnetwork-based model will have “holes” therein, this will result ininaccuracies within the mapped space. Since a neural network is anempirical model, it is only as accurate as the training data set. Eventhough the model may not have been trained upon a given set of inputs,it will extrapolate the output and predict a value given a set ofinputs, even though these inputs are mapped across a space that isquestionable. In these areas, the confidence level in the predictedoutput is relatively low. This is described in detail in U.S. patentapplication Ser. No. 08/025,184, filed Mar. 2, 1993, which isincorporated herein by reference.

[0118] Referring now to FIG. 17, there is illustrated a flowchartdepicting the statistical method for generating the filter and definingthe end point 186 in FIG. 15. The flowchart is initiated at a startblock 200 and then proceeds to a function block 202, wherein the controlvalues u(t+1) are calculated. However, prior to acquiring these controlvalues, the filtering operation must be a processed. The program willflow to a function block 204 to determine the accuracy of thecontroller. This is done off-line by analyzing the model predictedvalues compared to the actual values, and calculating the standarddeviation of the error in areas where the target is undisturbed. Themodel accuracy of e_(m)(t) is defined as follows:

e _(m)(t)=a(t)−p(t)  (026)

[0119] where:

[0120] e_(m)=model error,

[0121] a=actual value

[0122] p=model predicted value

[0123] The model accuracy is defined by the following equation:

Acc=H*σ _(m)  (027)

[0124] where:

[0125] Acc=accuracy in terms of minimal detector errorH = significance  level = 1  67%  confidence   = 2  95%  confidence   = 3  99.5%  confidence

[0126] σ_(m)=standard deviation of e_(m)(t).

[0127] The program then flows to a function block 206 to compare thecontroller error e_(c)(t) with the model accuracy. This is done bytaking the difference between the predicted value (measured value) andthe desired value. This is the controller error calculation as follows:

e _(c)(t)=d(t)−m(t)  (028)

[0128] where:

[0129] ec=controller error

[0130] d=desired value

[0131] m=measured value

[0132] The program will then flow to a decision block 208 to determineif the error is within the accuracy limits. The determination as towhether the error is within the accuracy limits is done utilizingShewhart limits. With this type of limit and this type of filter, adetermination is made as to whether the controller error e_(c)(t) meetsthe following conditions: e_(c)(t)≧−1*Acc and e_(c)(t)≦+1*Ace, theneither the control action is suppressed or not suppressed. If it iswithin the accuracy limits, then the control action is suppressed andthe program flows along a “Y” path. If not, the program will flow alongthe “N” path to function block 210 to accept the u(t+1) values. If theerror lies within the controller accuracy, then the program flows alongthe “Y” path from decision block 208 to a function block 212 tocalculate the running accumulation of errors. This is formed utilizing aCUSUM approach. The controller CUSUM calculations are done as follows:

S _(low)=min(0,S _(low)(t−1)+d(t)−m(t))−Σ(m)+k)  (029)

S _(hi)=max(0,S _(hi)(t−1)+[d(t)−m(t))−Σ(m)]−k)  (030)

[0133] where:

[0134] S_(hi)=Running Positive Qsum

[0135] S_(low)=Running Negative Qsum

[0136] k=Tuning factor—minimal detectable change threshold

[0137] with the following defined:

[0138] Hq=significance level. Values of (j,k) can be found so that theCUSUM control chart will have significance levels equivalent to Shewhartcontrol charts.

[0139] The program will then flow to a decision block 214 to determineif the CUSUM limits check out, i.e., it will determine if the Qsumvalues are within the limits. If the Qsum, the accumulated sum error, iswithin the established limits, the program will then flow along the “Y”path. And, if it is not within the limits, it will flow along the “N”path to accept the controller values u(t+1). The limits are determinedif both the value of S_(hi)≧+1*Hq and S_(low)≦−1*Hq. Both of theseactions will result in this program flowing along the “Y” path. If itflows along the “N” path, the sum is set equal to zero and then theprogram flows to the function block 210. If the Qsum values are withinthe limits, it flows along the “Y” path to a function block 218 whereina determination is made as to whether the user wishes to perturb theprocess. If so, the program will flow along the “Y” path to the functionblock 210 to accept the control values u(t+1). If not, the program willflow along the “N” path from decision block 218 to a function block 222to suppress the controller values u(t+1). The decision block 218, whenit flows along the “Y” path, is a process that allows the user tore-identify the model for on-line adaptation, i.e., retrain the model.This is for the purpose of data collection and once the data has beencollected, the system is then reactivated.

[0140] Referring now to FIG. 18, there is illustrated a block diagram ofthe overall optimization procedure. In the first step of the procedure,the initial steady-state values {Y_(ss) ^(i), U_(ss) ^(i)} and the finalsteady-state values {Y_(ss) ^(f), U_(ss) ^(f)} are determined, asdefined in blocks 226 and 228, respectively. In some calculations, boththe initial and the final steady-state values are required. The initialsteady-state values are utilized to define the coefficients a^(i), b^(i)in a block 228. As described above, this utilizes the coefficientscaling of the b-coefficients. Similarly, the steady-state values inblock 228 are utilized to define the coefficients a^(f), b^(f), it beingnoted that only the b-coefficients are also defined in a block 229. Oncethe beginning and end points are defined, it is then necessary todetermine the path therebetween. This is provided by block 230 for pathoptimization. There are two methods for determining how the dynamiccontroller traverses this path. The first, as described above, is todefine the approximate dynamic gain over the path from the initial gainto the final gain. As noted above, this can incur some instabilities.The second method is to define the input values over the horizon fromthe initial value to the final value such that the desired value Y_(ss)^(f) is achieved. Thereafter, the gain can be set for the dynamic modelby scaling the b-coefficients. As noted above, this second method doesnot necessarily force the predicted value of the output y^(p)(t) along adefined path; rather, it defines the characteristics of the model as afunction of the error between the predicted and actual values over thehorizon from the initial value to the final or desired value. Thiseffectively defines the input values for each point on the trajectoryor, alternatively, the dynamic gain along the trajectory.

[0141] Referring now to FIG. 18a, there is illustrated a diagrammaticrepresentation of the manner in which the path is mapped through theinput and output space. The steady-state model is operable to predictboth the output steady-state value Y_(ss) ^(i) at a value of k=0, theinitial steady-state value, and the output steady-state value Y_(ss)^(i) at a time t+N where k=N, the final steady-state value. At theinitial steady-state value, there is defined a region 227, which region227 comprises a surface in the output space in the proximity of theinitial steady-state value, which initial steady-state value also liesin the output space. This defines the range over which the dynamiccontroller can operate and the range over which it is valid. At thefinal steady-state value, if the gain were not changed, the dynamicmodel would not be valid. However, by utilizing the steady-state modelto calculate the steady-state gain at the final steady-state value andthen force the gain of the dynamic model to equal that of thesteady-state model, the dynamic model then becomes valid over a region229, proximate the final steady-state value. This is at a value of k=N.The problem that arises is how to define the path between the initialand final steady-state values. One possibility, as mentionedhereinabove, is to utilize the steady-state model to calculate thesteady-state gain at multiple points along the path between the initialsteady-state value and the final steady-state value and then define thedynamic gain at those points. This could be utilized in an optimizationroutine, which could require a large number of calculations. If thecomputational ability were there, this would provide a continuouscalculation for the dynamic gain along the path traversed between theinitial steady-state value and the final steady-state value utilizingthe steady-state gain. However, it is possible that the steady-statemodel is not valid in regions between the initial and final steady-statevalues, i.e., there is a low confidence level due to the fact that thetraining in those regions may not be adequate to define the modeltherein. Therefore, the dynamic gain is approximated in these regions,the primary goal being to have some adjustment of the dynamic modelalong the path between the initial and the final steady-state valuesduring the optimization procedure. This allows the dynamic operation ofthe model to be defined. This is represented by a number of surfaces 225as shown in phantom.

[0142] Referring now to FIG. 19, there is illustrated a flow chartdepicting the optimization algorithm. The program is initiated at astart block 232 and then proceeds to a function block 234 to define theactual input values u^(a)(t) at the beginning of the horizon, thistypically being the steady-state value U_(ss). The program then flows toa function block 235 to generate the predicted values y^(p)(k) over thehorizon for all k for the fixed input values. The program then flows toa function block 236 to generate the error E(k) over the horizon for allk for the previously generated y^(p)(k). These errors and the predictedvalues are then accumulated, as noted by function block 238. The programthen flows to a function block 240 to optimize the value of u(t) foreach value of k in one embodiment. This will result in k-values foru(t). Of course, it is sufficient to utilize less calculations than thetotal k-calculations over the horizon to provide for a more efficientalgorithm. The results of this optimization will provide the predictedchange Δu(t+k) for each value of k in a function block 242. The programthen flows to a function block 243 wherein the value of u(t+k) for eachu will be incremented by the value Δu(t+k). The program will then flowto a decision block 244 to determine if the objective function notedabove is less than or equal to a desired value. If not, the program willflow back along an “N” path to the input of function block 235 to againmake another pass. This operation was described above with respect toFIGS. 11a and 11 b. When the objective function is in an acceptablelevel, the program will flow from decision block 244 along the “Y” pathto a function block 245 to set the value of u(t+k) for all u. Thisdefines the path. The program then flows to an End block 246.

[0143] Steady State Gain Determination

[0144] Referring now to FIG. 20, there is illustrated a plot of theinput space and the error associated therewith. The input space iscomprised of two variables x₁ and x₂. The y-axis represents the functionf(x₁, x₂). In the plane of x₁ and x₂, there is illustrated a region 250,which represents the training data set. Areas outside of the region 250constitute regions of no data, i.e., a low confidence level region. Thefunction Y will have an error associated therewith. This is representedby a plane 252. However, the error in the plane 250 is only valid in aregion 254, which corresponds to the region 250. Areas outside of region254 on plane 252 have an unknown error associated therewith. As aresult, whenever the network is operated outside of the region 250 withthe error region 254, the confidence level in the network is low. Ofcourse, the confidence level will not abruptly change once outside ofthe known data regions but, rather, decreases as the distance from theknown data in the training set increases. This is represented in FIG. 21wherein the confidence is defined as α(x). It can be seen from FIG. 21that the confidence level α(x) is high in regions overlying the region250.

[0145] Once the system is operating outside of the training dataregions, i.e., in a low confidence region, the accuracy of the neuralnet is relatively low. In accordance with one aspect of the preferredembodiment, a first principles model g(x) is utilized to governsteady-state operation. The switching between the neural network modelf(x) and the first principle models g(x) is not an abrupt switching but,rather, it is a mixture of the two.

[0146] The steady-state gain relationship is defined in Equation 7 andis set forth in a more simple manner as follows: $\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {f\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (031)\end{matrix}$

[0147] A new output function Y(u) is defined to take into account theconfidence factor α(u) as follows:

Y({right arrow over (u)})=α({right arrow over (u)}).f({right arrow over(u)})+(1−α({right arrow over (u)}))g({right arrow over (u)})  (032)

[0148] where:

[0149] α(u)=confidence in model f(u)

[0150] α(u) in the range of 0→1

[0151] α(u)ε{0,1}

[0152] This will give rise to the relationship: $\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {Y\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (033)\end{matrix}$

[0153] In calculating the steady-state gain in accordance with thisEquation utilizing the output relationship Y(u), the following willresult: $\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = {{\frac{\partial\left( {\alpha \left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)} \times {F\left( \overset{\rightarrow}{u} \right)}} + {{\alpha \left( \overset{\rightarrow}{u} \right)}\frac{\partial\left( {F\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} + {\frac{\partial\left( {1 - {\alpha \left( \overset{\rightarrow}{u} \right)}} \right)}{\partial\left( \overset{\rightarrow}{u} \right)} \times {g\left( \overset{\rightarrow}{u} \right)}} + {\left( {1 - {\alpha \left( \overset{\rightarrow}{u} \right)}} \right)\frac{\partial\left( {g\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}}}} & (034)\end{matrix}$

[0154] Referring now to FIG. 22, there is illustrated a block diagram ofthe embodiment for realizing the switching between the neural networkmodel and the first principles model. A neural network block 300 isprovided for the function f(u), a first principle block 302 is providedfor the function g(u) and a confidence level block 304 for the functionα(u). The input u(t) is input to each of the blocks 300-304. The outputof block 304 is processed through a subtraction block 306 to generatethe function 1-α(u), which is input to a multiplication block 308 formultiplication with the output of the first principles block 302. Thisprovides the function (1−α(u))*g(u). Additionally, the output of theconfidence block 304 is input to a multiplication block 310 formultiplication with the output of the neural network block 300. Thisprovides the function f(u)*α(u). The output of block 308 and the outputof block 310 are input to a summation block 312 to provide the outputY(u).

[0155] Referring now to FIG. 23, there is illustrated an alternateembodiment which utilizes discreet switching. The output of the firstprinciples block 302 and the neural network block 300 are provided andare operable to receive the input x(t). The output of the network block300 and first principles block 302 are input to a switch 320, the switch320 operable to select either the output of the first principals block302 or the output of the neural network block 300. The output of theswitch 320 provides the output Y(u).

[0156] The switch 320 is controlled by a domain analyzer 322. The domainanalyzer 322 is operable to receive the input x(t) and determine whetherthe domain is one that is within a valid region of the network 300. Ifnot, the switch 320 is controlled to utilize the first principlesoperation in the first principles block 302. The domain analyzer 322utilizes the training database 326 to determine the regions in which thetraining data is valid for the network 300. Alternatively, the domainanalyzer 320 could utilize the confidence factor α(u) and compare thiswith a threshold, below which the first principles model 302 would beutilized.

[0157] Non-Linear Mill Control

[0158] Overall, model predictive control (MPC) has been the standardsupervisory control tool for such processes as are required in thecement industry. In the cement industry, particulate is fabricated witha kiln/cooler to generate raw material and then to grind this materialwith a mill. The overall kiln/cooler application, in the presentembodiment, utilizes a model of the process rather than a model of theoperator. This model will provide continuous regulation and disturbancerejection which will allow the application to recover from major upsets,such as coating drop three times faster than typical operatorintervention.

[0159] In general, mills demonstrate a severe non-linear behavior. Thiscan present a problem in various aspects due to the fact that the gainsat different locations within the input space can change. The cementkilns and coolers present a very difficult problem, in that theassociated processes, both chemical and physical, are in theory simple,but in practice complex. This is especially so when commercial issuessuch as quality and costs of production are considered. Themanufacturing of cement, and its primary ingredient, clinker, has anumber of conflicting control objectives, which are to maximizeproduction, minimize costs, and maximize efficiency, while at the sametime maintaining minimum quality specifications. All of thisoptimization must take place within various environmental, thermodynamicand mechanical constraints.

[0160] A primary technique of control for clinker has been the operator.As rotary cement kilns and automation technology evolve, variousautomation solutions have been developed for the cement industry. Thesesolutions have been successful to a greater or lessor extent. In thepresent application, the process is modeled, rather than the operator,and model predictive control is utilized. Moves are made every controlcycle to the process based on continuous feedback of key measurements.This gives rise to a continuous MPC action, as opposed to theintermittent, albeit frequent moves made by the typical expert system.In addition, as will be described hereinbelow, the approach describedutilizes full multivariable control (MVC) techniques, which take intoaccount all coupled interactions in the kiln/cooler process.

[0161] The cement mill is utilized to manufacture the various grades ofcement after processing of the raw material, which are defined by theirchemical composition and fineness (particle size distribution). Thecontrol objectives are thus to maximize production at minimum cost,i.e., low energy consumption for the various product grades, chemicalcompositions and specified fineness. In general, the mill utilizes aclosed circuit where separators in the feed-back are utilized toclassify the mill output into oversized and undersized product streams.The oversized stream, which does not conform to specification requiredfor correct cement strength, is fed back into the mill for furthergrinding and size reduction. Depending upon the type of mill, controlsinclude fresh feed, recirculating-load, as well as separator speed, allof which are used by the operator to control fineness, energyconsumption and throughput.

[0162] In general, the mill grinding equations take the form of:

ln(P)=k ₁ +k ₂ *F

[0163] where:

[0164] P=particle size

[0165] F=feed rate

[0166] k₁ and k₂ are constants.

[0167] It has generally been stated in the literature that grindingmodel equations are non-linear and hence, direct application of linearcontrol theory is not possible. The primary reason for this is that theoperation of the plant is only non-linear in very small regions of theinput space. Once the process has traversed, i.e., “stepped,” from oneportion of the input space to another portion thereof, the overall modelchanges, i.e., it is non-linear. This lends itself to non-linearmodeling techniques. However, most control systems are linear in nature,especially those that model the dynamics of the plant.

[0168] Referring now to FIG. 24, there is illustrated a diagrammaticview of the kiln/cooler configuration and the selected instrumentationutilized for optimal MPC control. This kiln/cooler consists of afive-stage suspension pre-heater kiln, with back-end firing(approximately fifteen percent of total firing). The cooler is a gratetype with a conversion upgrade on the first section. It has on-lineanalyzers for NOx, O₂ and CO located at the top of a preheater 2402which receives raw meal therein. The output of the preheater is input toa kiln 2404. There is provided a coal feed on an input 2406, the feedend and a coal feed 2408 on the firing end. The output of the kiln isinput to a cooler 2410 which has an input cooler fan 2412. The output ofthe cooler provides the clinker. The overall plan is fully instrumentedwith all necessary measurements, e.g., temperature, pressure and flowmeasurements such as coal, raw meal and grate air. The quality of theclinker production is maintained by the analysis of hourly samples ofthe raw meal feed, clinker and coal. This is supported by asemi-automated sampling system, and various modem laboratoryinfrastructure.

[0169] This system is controlled with an MPC controller (not shown) thatconsists of the MPC control engine as described hereinabove, as well asa real-time expert system that performs a variety of pre and postprocessing of control signals as well as various other functions such asMPC engine control, noise filtering, bias compensation and real-timetrending. This system will also perform set point tracking forbumperless transfer, and adaptive target selection. This allows for thecontroller tuning parameters to be changed according to various businessand/or process strategies.

[0170] The MPC is defined in two primary phases the first being themodeling phase in which the models of the kiln processes are developed.The second phase is the deployment phase, where the models arecommissioned and refined to a point where satisfactory control can beobtained. Central to commissioning is the tuning where the controller istweaked to provide the desired control and optimization. For examplethis could be: maximum production at the expense of quality, or optimalquality at the expense of efficiency.

[0171] The MPC models are developed from the analysis of test andprocess data, together with knowledge from the plant operators and otherdomain experts. The result is a matrix of time responses, where eachresponse reflects the dynamic interaction of a controlled variable to amanipulated variable.

[0172] The tuning involves the selection of targets (setpoints),weighting factors and various constraints for each variable. Thisdetermines how the controller will solve the control problem at anygiven time. The control of the kiln and its optimization within theabove set of constraints is solved every control cycle.

[0173] The solution chosen in a particular control cycle may not seem tobe necessarily optimal at that given time, but will be optimal withinthe solution space which has temporal as well as spatial dimensions.Thus the control solution is a series of trajectories into the future,where the whole solution is optimized with time. The very nature ofoptimal control in real time does not allow for a guarantee of a globaloptimal solution. However the calculation of an optimal solution withina finite amount of time is itself a class of optimization.

[0174] Some of the tuning parameters, which can be changed duringoperations, include:

[0175] 1) Targets. Targets can be set for both controlled andmanipulated variables, and the MPC controller will try and force allvariables to their desired targets. In particular setting a target for amanipulated variable such as coal allows for optimization, and inparticular efficiency, because the controller will continually seek alower coal flow while maintaining production and quality. For somevariables such as O₂, a target may not be necessary, and the variableswill be allowed to roam within a band of constraints.

[0176] 2) Priorities. Setting relative priorities between manipulatedvariables and controlled variables allows the controller to prioritizewhich are more important problems to solve, and what type of solution toapply. Underspecified multivariable control (more manipulated variablesthan controlled variables, as is the case in this application) impliesthat for every problem there will be more than one solution, but withinconstraints one solution will generally be more optimal than others. Forexample, too high a hood temperature can be controlled by, (a) reducingfuel, (b) increasing the grate speed, or (c) increasing cooler airflow,or a combination of the above.

[0177] 3) Hard Constraints. Setting upper and lower hard constraints foreach process variable, for example, minimum and maximum grate speed.These values which are usually defined by the mechanical and operationallimitations of the grate. Maintaining these constraints is obviouslyrealizable with controlled variables such as ID-fan speed, but is moredifficult to achieve with, for example, hood temperature. However whenhood temperature exceeds a upper hard constraint of say 1200° C., thecontroller will switch priority to this temperature excursion, and allother control problems will “take a back seat” to the solution requiredto bring this temperature back into the allowable operating zone.

[0178] 4) Soft upper and lower constraints. If any process variablepenetrates into the soft constraint area, penalties will be incurredthat will begin to prioritize the solution of this problem. Continuouspenetration into this area will cause increasing prioritization of thisproblem, thus in effect creating an adaptive prioritization, whichchanges with the plant state.

[0179] 5) Maximum rate of change constraints. These parameters are onlyapplicable to the manipulated variables, and generally reflect amechanical of physical limitation of the equipment used, for examplemaximum coal feed rate.

[0180] From a clinker production point of view the functions of the MPCapplication can be viewed as follows:

[0181] 1) Kiln Combustion Control where manipulated variables such asID-fan speed and fuel-flow are manipulated to control primarily O₂. WhenCO rises above a specified threshold constraint, it will override andbecome the controlled variable. The controller is tuned to heavilypenalize high CO values, steer the CO back into an acceptable operatingregion, and rapidly return to O₂ control.

[0182] 2) Kiln Thermal “Hinge Point” Control adjusts total coal, coolergrate speed, and cooler fans to control the hood temperature. The hoodtemperature is conceptualized as the “hinge” point on which the kilntemperature profile hangs. The controller is tuned to constantlyminimize cooler grate speed and cooler fans, so that heat recovery fromthe cooler is maximized, while minimizing moves to coal.

[0183] 3) Kiln Thermal “Swing Arm” Control adjusts percent coal to thekiln backend, in order to control clinker free lime based on hourly labfeedback. This control function is about three times slower than thehinge point control, which maintains hood temperature at a fixed target.The “swing arm effect” raises or lowers the back end temperature with aconstant firing end temperature to compensate for changes in free lime.This is in effect part of the quality control.

[0184] Kiln combustion control, kiln thermal hinge point control, andkiln thermal swing arm control are implemented in a single MPCcontroller. Kiln speed is included as a disturbance variable, as theproduction philosophy, in one embodiment, calls for setting a productionrate to meet various commercial obligations. This means that any changesto kiln speed and hence production rate by the operator will be takeninto account in the MPC predictions, but the MPC controller will not beable to move kiln speed.

[0185] The control system allows for customization of the interfacebetween the plant and the MPC special control functions, the specialcontrol functions implemented including:

[0186] 1) Total Coal Control allows the operator to enter a total coalor fuel flow setpoint. The control system “wrapper” splits the move tothe front and back individual coal flow controllers while maintainingthe percent of coal to the back constant. The purpose of this controlfunction is to allow heating and cooling of the kiln while maintaining aconstant energy profile from the preheaters through to the firing end ofthe kiln. This provides a solid basis for the temperature “hinge point”advanced control function previously described.

[0187] 2) Percent Coal to the Back Control allows the operator to entera percent coal to the back target and implements the moves to the frontand back coal flow controllers to enforce it. The purpose of thiscontrol is to allow the back end temperature to be swung up or down bythe thermal “swing arm” advanced control function.

[0188] 3) Feed-to-Speed Ratio Control adjusts raw meal feed to the kilnto maintain a constant ratio to kiln speed. The purpose of thiscontroller is to maintain a constant bed depth in the kiln, which isimportant for long-term stabilization.

[0189] 4) Cooler Fans Control is a move splitter that relates a singlegeneric cooler air fans setpoint to actual setpoints required by ncooler air fans. The expert system wrapper through intelligentdiagnostics or by operator selection can determine which of the aircooler fans will be placed under direct control of the MPC controller,thus allowing for full control irrespective of the (for example)maintenance being undertaken on any fans.

[0190] 5) Gas analyzer selection. The control system automatically scansthe health of the gas analyzers, and will switch to the alternativeanalyzer should the signals become “unhealthily’. In addition thecontrol system is used to intelligently extract the fundamental controlsignals from the O₂ and CO readings, which are badly distorted by purgespikes etc.

[0191] Referring now to FIG. 25, there is illustrated a block diagram ofthe non-linear mill and the basic instrumentation utilized for advancedcontrol therein. The particle size overall is measured as “Blaine” incm²/gm , and is controlled by the operator through adjustment of thefresh feed rate and the separator speed. The mill is a ball mill, whichis referred to by reference numeral 2502. The fresh feed is metered by afresh feed input device 2506 which receives mined or processed materialinto the mill 2502, which mill is a mechanical device that grinds thismined or processed material. In this embodiment, the mill is a ballmill, which is a large cylinder that is loaded with steel balls. Themill 2502 rotates and is controlled by a motor 2508 and, as the materialpasses therethrough, it is comminuted to a specified fineness or Blaine.The output of the mill is input to an elevator 2510 which receives theoutput of the mill 2502 and inputs it to a separator 2512. This is afeedback system which is different than an “open circuit” mill and isreferred to as a “closed circuit” mill. The mill product is fed into theseparator 2512, which separator 2512 then divides the product into twostreams. The particles that meet product specification are allowed toexit the system as an output, represented by a reference numeral 2514,whereas the particles that are too large are fed back to the input ofthe mill 2502 through a return 2516 referred to as the course return.

[0192] There are provided various sensors for the operation of the mill.The separator speed is controlled by an input 2518 which signal isgenerated by a controller 2520. The elevator 2510 provides an output2522, which constitutes basically the current required by the elevator2510. This can be correlated to the output of the mill, as the largerthe output, the more current that is required to lift it to theseparator 2512. Additionally, the motor 2508 can provide an output.There is additionally provided an “ear,” which is a sonic device thatmonitors the operation of the mill through various sonic techniques. Itis known that the operation of the mill can be audibly detected suchthat operation within certain frequency ranges indicates that the millis running well and in other frequency ranges that it is not runningwell, i.e., it is not optimum.

[0193] Overall, the mill-separator-return system is referred to as a“mill circuit.”The main control variable for a mill circuit is productparticle size, the output, and fresh feed is manipulated to control thisvariable. A secondary control variable is return and separator speed ismanipulated to control this variable. There are also provided variousconstants as inputs and constraints for the control operation. Thiscontroller 2520 will also control fresh feed on a line 2524.

[0194] The response of particle size to a move in fresh feed is known tobe slow (one-two hours) and is dominated by dead time. Where a dead timeto time constant ratio exceeding 0.5 is known to be difficult to controlwithout model predictive control techniques, documents ratios for theresponse of particle size to a move in fresh feed includes 0.9 and 1.3.

[0195] In the case of a closed-circuit mill, a move to fresh feedeffects not only the product particle size, but also the return flow.Also, a move to separator speed effects not only the return flow, butalso the particle size. This is a fully interactive multi-variablecontrol problem.

[0196] The controller adjusts fresh feed and separator speed to controlBlaine and return. It also includes motor and sonic ear as outputs, andthe sonic ear is currently used as a constraint variable. That meanswhen the sonic ear decibel reading is too high then fresh feed isdecreased. In this way the controller maximizes feed to the sonic ear(choking) constraint.

[0197] Referring now to FIG. 26, there is illustrated a dynamic modelmatrix for the fresh feed and the separator speed for the measuredvariables of the Blaine Return Ear and motor. It can be seen that eachof these outputs has a minimum and maximum gain associated therewithdead-time delay and various time constants.

[0198] Referring now to FIG. 27, there is illustrated a plot of logsheet data utilized to generate a gain model for the controller 2520.This constitutes the historical data utilized to train the steady statenon-linear model.

[0199] In general, the operation described hereinabove utilizes anon-linear controller which provides a model of the dynamics of theplants in a particular region. The only difference in the non-linearmodel between one region of the input space to a second region of theinput space is that associated with the dynamic gain “k.” This dynamicgain varies as the input space is traversed, i.e., the model is onlyvalid over a small region of the input space for a given dynamic gain.In order to compensate for this dynamic gain of a dynamic linear model,i.e., the controller, a non-linear steady state model of the overallprocess is utilized to calculate a steady-state gain “K” which is thenutilized to modify the dynamic gain “k.” This was described in detailhereinabove. In order to utilize this model, it is necessary to firstmodel the non-linear operation of the process, i.e., determining anon-linear steady state model, and then also determine the variousdynamics of the system through “step testing.” The historical dataprovided by the log sheets of FIG. 27 provide this information which canthen be utilized to train the non-linear steady state model.

[0200] Referring now to FIGS. 28a and 28 b, there are illustrated plotsof process gains for the overall non-linear mill model with the scalesaltered. In FIG. 28a, there is illustrated a plot of the sensitivityversus the percent move of the Blaine. There are provided two plots, onefor separator speed and one for fresh feed. The upper plot illustratesthat the sensitivity of Blaine to separator speed is very low, whereasthe gain of the percent movement of fresh feed with respect to theBlaine varies considerably. It can be seen that at a minimum, the gainis approximately −1.0 and then approaches 0.0 as Blaine increases. Inthe plot of FIG. 28b, there are illustrated two plots for thesensitivity versus the percent return. In the upper plot, thatassociated with the percent movement of separator speed, it can be seenthat the gain varies from a maximum of 0.4 at a low return to a gain of0.0 at a high return value. The fresh feed varies similarly, with lessrange. In general, the plots of FIGS. 28a and 28 a, it can be seen thatthe sensitivity of the control variable, e.g., Blaine in FIG. 28a andreturn in the plot of FIG. 28b, as compared to the manipulated variablesof separator speed and fresh feed. The manipulated variables areillustrated as ranging from their minimum to maximum values.

[0201] In operation of the controller 2520, the process gains(sensitivities) are calculated from the neural network model (steadystate model) at each control execution, and downloaded into thepredictive models in the linear controller (dynamic model). Thisessentially provides the linear controller with the dynamic gains thatfollow the sensitivities exhibited by the steady-state behavior of theprocess, and thereby provides non-linear mill control utilizing a linearcontroller. With such a technique, the system can now operate over amuch broader specification without changing the tuning parameters or thepredictive models in the controller.

[0202] Referring now to FIG. 29, there is illustrated a plot ofnon-linear gains for fresh feed responses for both manipulatiblevariables and control variables. The manipulatible variable for thefresh feed is stepped from a value of 120.0 to approximately 70.0. Itcan be seen that the corresponding Blaine output and the return output,the controlled variables, also steps accordingly. However, it can beseen that a step size of 10.0 in the fresh feed input does not result inidentical step in either the Blaine or the return over the subsequentsteps. Therefore, it can be seen that this is clearly a non-linearsystem with varying gains.

[0203] Referring now to FIG. 30, there is illustrated four plots of thenon-linear gains for separator speed responses, wherein the separatorspeed as a manipulatible variable is stepped from a value ofapproximately 45.0 to 85.0, in steps of approximately 10.0. It can beseen that the return controlled variable makes an initial change with aninitial step that is fairly large compared to any change at the end. TheBlaine, by comparison, appears to change an identical amount each time.However, it can be seen that the response of the return with respect tochanges in the separator speed will result in a very non-linear system.

[0204] Referring now to FIG. 31, there is illustrated an overall outputfor the closed loop control operation wherein a target change for theBlaine is provided. In this system, it can be seen that the fresh feedis stepped at a time 3102 with a defined trajectory and this results inthe Blaine falling and the return rising and falling. Further, theseparator speed is also changed from one value to a lower value.However, the manipulatible variables have the trajectory thereof definedby the non-linear controller such that there is an initial point and afinal point and a trajectory therebetween. By correctly defining thistrajectory between the initial point and the final point, the trajectoryof the Blaine and the return can be predicted. This is due to the factthat the non-linear model models the dynamics of the plant, thesedynamics being learned from the step tests on the plant. This, ofcourse, is learned at one location in the input space and, by utilizingthe steady state gains from the non-linear steady state model tomanipulate the dynamic gains of the linear model, control can beeffected over different areas of the input space, even though theoperation is non-linear from one point in the input space to anotherspace.

[0205] Although the preferred embodiment has been described in detail,it should be understood that various changes, substitutions andalterations can be made therein without departing from the spirit andscope of the invention as defined by the appended claims.

What is claimed is:
 1. A method for controlling a non-linear plant,comprising the steps of: providing a linear controller having a lineargain k that is operable to receive inputs representing measuredvariables of the plant and predicting on an output of the linearcontroller predicted control values for manipulatible variables thatcontrol the plant; providing a non-linear model of the plant for storinga representation of the plant over a trained region of the operatinginput space and having a steady-state gain K associated therewith;adjusting the gain k of the linear model with the gain K of thenon-linear model in accordance with a predetermined relationship as themeasured variables change the operating region of the input space atwhich the linear controller is predicting the values for themanipulatible variables; and outputting the predictive manipulatiblevariables after the step of adjusting the gain k.
 2. The method of claim1, wherein the linear controller is operable to model the dynamics ofthe plant.
 3. The method of claim 1, wherein the dynamics of the plantare modeled over a defined region of the input space.
 4. The method ofclaim 1, wherein the training region of the operating input space overwhich the non-linear model is representative of the operation of theplant represents a space greater than that over which the linearcontroller is valid.
 5. The method of claim 1, and further comprisingthe step of controlling the operation of the plant with the predictedcontrol variables after the step of adjusting.